From: fortunati.luigi@gmail.com   
      
   Muscular father trains his son in tug of war   
   https://drive.google.com/file/d/1JHyDVrriwjUnlFoGUaHRFfpIsHBf4gY   
   /view?usp=sharing   
      
   The son pulls to the left with all his strength, while the father pulls   
   to the right just enough to keep the rope steady: the two opposing   
   forces balance each other, and the rope stays steady.   
      
   The father and son's feet are firmly planted on the ground, where   
   there's no stupid ice that could favor one or the other.   
      
   When the father wants his son to win, he decreases his strength and the   
   rope moves to the left; when he wants to pull his son toward him, he   
   increases his strength and the rope moves to the right.   
      
   How can the son's reaction be the same as the father's action when he   
   pulls him towards him (and the rope accelerates to the right) or when he   
   pretends to let the son win (and the rope accelerates to the left)?   
      
   Luigi Fortunati   
      
   [[Mod. note --   
   Several points:   
      
   First, the picture in the google drive shows the rope NOT being   
   horizontal, i.e., there are vertical components to the forces.   
   I'm going to simplify things by instead pretending that the rope   
   is horizontal (i.e., that the father lowers his hands to be at the   
   same height as the son's hands) so that we only need to consider   
   horizontal forces.   
      
   Second, we can also greatly simplifies things if we neglect any   
   stretching of the rope.  This lets us refer to "the acceleration   
   of the rope" without having to worry about different parts of the   
   rope having different accelerations.   
      
   With these simplifications, we can analyse the system relatively   
   easiy:  Let's start by listing the forces of interest:   
      F_father_on_rope = rightward pull of father on rope (> 0)   
      F_rope_on_father = leftward pull of rope on father (< 0)   
      F_son_on_rope = leftward pull of son on rope (< 0)   
      F_rope_on_son = rightward pull of rope on son (> 0)   
      
   The net force on the rope is just the difference between the   
   father's rightward pull on the rope and the son's leftward pull,   
   i.e.,   
      F_net_on_rope = F_father_on_rope + F_son_on_rope   
      
   Newton's 2nd law says that the rope accelerates (horizontally)   
   if and only if F_net_on_rope is nonzero, i.e., if and only if   
   F_father_on_rope and F_son_on_rope are different in magnitude.   
   More precisely, Newton's 2nd law says that   
      F_net_on_rope = F_father_on_rope + F_son_on_rope = m_rope a   
   where m_rope is the mass of the rope and /a/ is the (horizontal)   
   acceleration of the rope.   
      
   Newton's 3rd law only applies to bodies that directly exert forces   
   on each other.  In this case, the father directly exerts a force on   
   the rope, and vice versa, so Newton's 3rd law says that   
      F_rope_on_father = - F_father_on_rope   
   And, the son directly exerts a force on the rope, and vice versa, so   
   Newton's 3rd law says that   
      F_rope_on_son = - F_son_on_rope   
      
   Because the father and son don't directly exert forces on each other,   
   Newton's 3rd law says nothing about the relationship between   
   F_father_on_rope and F_son_on_rope.   
      
   Now, consider Luigi's question   
   > How can the son's reaction be the same as the father's action when he   
   > pulls him towards him (and the rope accelerates to the right) or when he   
   > pretends to let the son win (and the rope accelerates to the left)?   
      
   We need to be precise about which forces we're considering.   
   As I noted, Newton's 3rd law says nothing about the relationship   
   between F_father_on_rope and F_son_on_rope.  However, Newton's 2nd   
   law does relate those, as described above.   
   -- jt]]   
      
   --- SoupGate-Win32 v1.05   
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